, Volume 18, Issue 1, pp 15–35 | Cite as

Rice formula for processes with jumps and applications



We extend Rice Formula to a process which is the sum of two independent processes: a smooth process and a pure jump process with finitely many jumps. Formulas for the mean number of both continuous and discontinuous crossings through a fixed level on a compact time interval are obtained. We present examples in which we compute explicitly the mean number of crossings and compare which kind of crossings dominates for high levels. In one of the examples the leading term of the tail of the distribution function of the maximum of the process over a compact time interval as the level goes to infinity is obtained. We end giving a generalization, to the non-stationary case, of Borovkov-Last’s Rice Formula for Piecewise Deterministic Markov Processes.


Rice formula Level crossings Process with jumps Number of continuous and discontinuous crossings 

AMS 2000 Subject Classifications

Primary—60G10 stationary processes 60J75 Jump processes Secondary—60G70 Extreme value theory; extremal processes 


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© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Departamento de Matemática y Estadística del Litoral, Regional NorteUniversidad de la RepúblicaSaltoUruguay
  2. 2.Centro de Matemática, Facultad de CienciasUniversidad de la RepúblicaMontevideoUruguay

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